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Application of the Cutting Stock Problem to a Construction Company: A Case Study


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This paper presents an application of the well-known cutting stock problem to a construction firm. The goal of the 1Dimensional (1D) cutting stock problem is to cut the bars of desired lengths in required quantities from longer bars of given length. The company for which we carried out this study encounters 1D cutting stock problem in cutting steel bars (reinforcement bars) for its construction projects. We have developed several solution approaches to solving the company’s problem: Building and solving an integer programming (IP) model in a modeling environment, developing our own software that uses a mixed integer programming (MIP) software library, and testing some of the commercial software packages available on the internet. In this paper, we summarize our experiences with all the three approaches. We also present a benchmark of existing commercial software packages, and some critical insights. Finally, we suggest a visual approach for increasing performance in solving the cutting stock problem and demonstrate the applicability of this approach using the company’s data on two construction projects.

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Application of the Cutting Stock Problem to a Construction Company: A Case Study

  1. 1. Alp, S., Ertek, G. and Birbil, S. I. (2006). "Application of the cutting stock problem to aconstruction company: A case study.” 5th International Symposium on IntelligentManufacturing Systems, Sakarya, Turkey.Note: This is the final draft version of this paper. Please cite this paper (or this final draft) asabove. You can download this final draft from Application of the Cutting Stock Problem to a Construction Company: A Case Study Seda Alp, Gurdal Ertek, S.Ilker Birbil Sabanci University, Faculty of Engineering and Natural Sciences, Orhanli, Tuzla, 34956, Istanbul, Turkey Tel: +90(216)483-9568 Fax: +90(216)483-9550 1 / 18
  2. 2. AbstractThis paper presents an application of the well-known cutting stock problem to aconstruction firm. The goal of the 1Dimensional (1D) cutting stock problem is tocut the bars of desired lengths in required quantities from longer bars of givenlength. The company for which we carried out this study encounters 1D cuttingstock problem in cutting steel bars (reinforcement bars) for its constructionprojects. We have developed several solution approaches to solving the company’sproblem: Building and solving an integer programming (IP) model in a modelingenvironment, developing our own software that uses a mixed integerprogramming (MIP) software library, and testing some of the commercialsoftware packages available on the internet. In this paper, we summarize ourexperiences with all the three approaches. We also present a benchmark ofexisting commercial software packages, and some critical insights. Finally, wesuggest a visual approach for increasing performance in solving the cutting stockproblem and demonstrate the applicability of this approach using the company’sdata on two construction projects.Submission areas: Decision Support System, Management Information SystemsIntroductionThe cutting stock problem is encountered in many industries, including the constructionindustry. In constructions, steel bars (reinforcement bars) of certain lengths are needed inspecified quantities. These bars are cut from long steel bars (with lengths as much as 1200 cms)according to the cutting patterns. The problem is to find the "optimal" cutting patterns, wherethe total number of long steel bars used is minimized, subject to the constraint that the desiredshorter steel bars are cut in required quantities. 2 / 18
  3. 3. The Problem SettingThe CompanyMimag Makina Ltd, which is established in 1983, is a construction firm under the parentcompany, MIMAG Group1. Initial goal in founding the company group was to produce steelfabrication needs of the parent company, mainly formwork, scaffolding, and constructionequipment. Later the products were diversified, and found a general market among contractors.At present, the company designs, produces, supervises formwork and scaffolding systemssuitable for all kinds of structures such as dams, bridges, business centers, and industrial plants.Up to this date several projects were completed by Mimag, including housing projects (200residential sites), industrial installations, railway electrification, silos, chimneys, and variouscivil, mechanical and structural installation jobs are executed.Reinforced concrete, which is the combination of concrete and steel, is used in several projectsof Mimag because of the quality of durability and for economic purposes. One of the projectsthat Mimag had dealt with was the “Diesel/Kerosene Hydroprocessing and CCR ReformerProject” implemented in Izmit Refinery of Tupras, Turkey (Figure 1a). Tupras is the topcompany in Turkey with respect to sales revenues. Vertical and horizontal reinforcements werebuilt to replace some silos in the mentioned project. The other real data that we have used wasfrom the project “Waste Water Treatment Plant” conducted in Adana, Turkey. The project wasfor the reinforcement of slab. To determine the amount of steel bars used in reinforced concrete,several methods are used to minimize the amount of scrap. The standard steel bars used inreinforced concrete steel bars are 1200 cm and have various diameters ranging from 6mm to50mm (Figure 1b). These different steels with different lengths and diameters should be cut inan efficient manner such that the scrap is minimized (Figure 1c).1 3 / 18
  4. 4. (a) (b) (c)Figure 1. (a) 1200cm long steel bars at the Izmit - Tupras Construction site waiting to be cut. (b) The cutsteel bars with different lengths and diameters. (c) Equipment used to cut the steel bars. Source: 4 / 18
  5. 5. The 1D Cutting Stock ProblemThe mathematical model for the 1D cutting stock problem is presented below:SETS:I: set of patternsJ: set of lengthsDECISION VARIABLES:xi : number of bars cut according to pattern i.PARAMETERS:a ij : number of pieces of length j within one bar cut according to pattern ib j : required number of pieces of length jTHE MODEL min x i (1) iI s.t a ij xi  b j i , j  J (2) iI xi  0 , integer (3) 5 / 18
  6. 6. The goal of the model is to minimize the objective function (1), which consists of the totalnumber of long steel bars used. Constraint set (2) ensures that enough number of shorter barsare cut from long bars in required quantities. The decision variables are restricted to non-negative values (3).One can obtain the optimal solution to the cutting stock problem by using an IP/MIP solver. Forsolving large instances of the problem, the reader is referred to Gilmore and Gomory (1961).The Solution ApproachesSolving within the GAMS Modeling SystemThe first solution approach that we followed to tackle with Mimag’s 1D cutting stock problemwas building and solving an IP formulation within the GAMS Modeling System2. We haveimplemented our GAMS model based upon information provided in Kalvelagen (2003). Weinitially built the IP model for one of the subprojects within the Tupras Project, consisting of 8different lengths (Appendix A). One advantage of using GAMS is being able to represent thedata in matrix and vector forms. Another advantage is being assured that the optimal solution isfound. However, there are major drawbacks of this approach:  The patterns cannot be generated by the GAMS modeling language; they have to be generated through a separate program written in a general purpose programming language, such as C++.  Adding a new pattern requires “hacking” into the GAMS model and adding a new row to the exact appropriate place. GAMS modeling system is very sensitive to incorrect syntax, and can give hard-interpret error messages when the required syntax is not followed.2 6 / 18
  7. 7. Developing an IP-based Optimization ProgramThe shortcomings of using a system such as GAMS have led us to development of an IP-basedoptimization program for solving the problem. We had written a C++ program to provide thepattern data to be input into the GAMS model. However, this early implementation wasemploying nested loops (one loop for each additional length) to generate the feasible patterns.Such an implementation required modification of the source code for every new subproject data,since each subprojects can have different number of lengths required. Yet another issue was thefact that GAMS is a commercial software, which requires purchase of a new license for eachinstallation. We clearly needed a free-of-charge open-source modeling system or softwarelibrary to use as the optimization engine.lp_solve3 is a free solver based on revised simplex algorithm for solving linear programming(LP) problems and the Branch-and-bound technique for solving IP and MIP problems. Eventough it is developed in ANSI C, it can be called from programs written in other programminglanguages, such as C++, Java, C#, and Visual Basic, as well. We have selected to code ourprogram in Java4, since one of the authors possesses proficiency in this language. We have useda “Java wrapper” that enables the communication of the Java code with the lp_solveoptimization engine, and carried out our code development within the Eclipse IDE5 (IntegratedDevelopment Environment).Our Java program reads data from a text file, generates patterns efficiently through an arraybased representation of patterns, calls lp_solve to build and solve the IP model, and prints outthe solution. Currently, our program has minimal GUI (Graphical User Interface), since GUI canconsume great amounts of CPU time, and slow down the time to solve the cutting stockproblem. The bottleneck of our current implementation is the generation of the patterns takingup to 1 minute for small-size instances with up to 17 lengths. One larger-size instance with 25lengths took more than 10 minutes to solve. One possible solution to this problem is cuttingsome of the lengths from the next longer length. We utilized this approach for solving thementioned instance.3 htpp://java.sun.com5 7 / 18
  8. 8. We validated the correctness of our program by comparing its solution to the GAMS solution forthree different instances.Using Commercial SoftwareOur next approach to solving the Mimag’s problem was to test commercial 1D cutting stocksoftware available on the internet. The software packages were searched using the key word “1dimensional cutting stock program” at The first ~30 pages of the searchresults were scanned to retrieve applicable software. Only seven packages were found to havedemo versions available that can handle the largest instance (with 25 lengths) that we had.These packages and their web sites are listed in Appendix B.Benchmarking ResultsThe Performance DataSome important questions that we had in mind -upon completion of our Java program- were thefollowing; Q1. How effective is our program compared to the commercial packages that we downloaded? Q2. How do the commercial packages compared amongst them? Q3. Would it be considerably more effective to use two different software packages at the same time for each subproject and implement the better of the two solutions that they provide? Q4. Does batching of two or more subprojects and determining the best solution for the batch increase performance (decrease the cost)? 8 / 18
  9. 9. We have found the answers to the first three questions based upon Mimag’s data. For answeringthe fourth question, we propose a visual approach that requires drawing a chart to determinewhich subprojects to merge. We present our visual approach in the next section.We have carried out a performance benchmark of our program and the commercial packages byusing the data sets from Izmit (Tupras) and Adana projects of Mimag. The total number of barsused in our program’s solution and the commercial packages’ solutions are shown in decreasingorder in Table 1. The software marked with a (*) is the one which Mimag was planning topurchase.Table 1. The total number of bars found by the benchmarked softwareSoftware Name Total number of BarsJava Program 10399Software 1 10400Software 2 10402Software 3 10403Software 4 10410Software 5* 10419Software 6 10577Software 7 11858InsightsAs can be seen from Table 1, our Java program outperformance all the commercial packages.This is naturally expected, since we know from optimization theory that IP solution gives the 9 / 18
  10. 10. optimal objective function value. The next thing that we noticed is that none of the commercialpackages have achieved the optimal total number of bars. This indicates that the softwarepackages are not implementing an optimal algorithm, but are implementing heuristictechniques.We noticed that the first five software packages give solutions very close to the optimal. So aconstruction company can select any of these five software packages, based on other criteriasuch as price, on-line support, usability, data import/export capabilities, and reporting quality(including visualization of optimal cutting patterns and their quantities). One very importantobservation is that Software 6 and especially Software 7 perform poorly. A company shoulddefinitely avoid using Software 7, which also happens to be the most expensive package amongstall. Software 7 is distinguished from all others by its capability to import/export to and from MSExcel. Meanwhile, Software 6 has the best GUI amongst all packages. Thus, the softwareselection should be based on performance tests, rather than price, data import/exportcapabilities, or GUI. We have so far answered Q1 and Q2. Since the software packages thatperform well are extremely close to the optimal there is no need to use two of those softwarepackages, thus answering Q3.A Visual Approach for Determining How to Batch SubprojectsWe now present a visual approach that enables a decision maker to select which subproject-diameter combinations can be batched (combined) to be cut simultaneously. This approach is ananswer to Q4 posed in the previous section. We project that the approach we describe below canreduce the number of long bars cut (which is the objective function of 1D cutting stock problem),and suggest that it can be implemented in cutting stock software packages.Figure 2 shows the chart that we propose6, where each point represents a subproject-diametercombination. The x-axis shows values of “Optimal number of bars used”, the y-axis shows valuesof “Diameters” (in cm), and the color of the points shows “Optimal usage” (as a percentage) forthat subproject, where dark colors denote lower usages (bad performance). To generate this6 The chart is generated using Miner3D Software ( 10 / 18
  11. 11. chart, the cutting stock problem has to be solved for each subproject, and the three values thatwill be displayed in the chart have to be calculated. This is a fairly easy thing to do, since steelbar requirements for each subproject of a project is determined apriori and stays fixed throughout the construction project.The approach we suggest is the following: 1. Start with the highest “Diameter” value on the y-axis and examine the points that reside on that row. We start searching from the biggest diameter since the cost of steel bars is depended upon the weight, which is a quadratic function of the diameter. 2. If there is a point with a dark color on that row which has a large “Optimal number of bars used” value on the x-axis, then consider batching that with another point (subproject-diameter combination). For example in Figure 2, we first consider batching points A and B. 3. Test using software whether the batching possibility in Step 2 brings any improvements. If batching brings significant improvements go ahead and carry out the batching while cutting the steel bars. If batching does not bring significant improvement carry out the cutting separately. 4. Return back to Step 1, considering the next smaller diameter, until you cannot find any batching possibilities that could bring significant savings. For example, the next batching possibility in Figure 2 is combining points C and D.When we applied the proposed visual approach, we observed that the two considered projects(Izmit and Adana) did not include great improvement opportunities through batching. We evenobserved that batching could result in a worse performance. Therefore, the decision makershould be aware that batching is not always a good option. Even though we have not gainedmeaningful improvements by batching, when using our data, we believe that for other projectsthere can be significant savings. One could especially expect good opportunities when there area large number of projects/subprojects managed simultaneously, and a large number of steelbars to be cut. We suggest carrying out more extensive tests as a future research area. 11 / 18
  12. 12. One important issue in batching is the additional inventory of cut steel bars that will have to bestored. This translates into additional inventory holding costs, which are incurred due to timevalue of money. Any analysis of batching should also consider this additional cost. Figure 2. The chart that displays performance data on subprojects.ConclusionsMotivated by a real world problem encountered by a construction firm, we have tested andbenchmarked commercial software packages for solving the 1D cutting stock problem. Ourfindings suggest that available software packages should be tested with sample project databefore adoption. We observed that software packages with the best import/export capabilitiesand/or GUI can perform very poorly. Besides, a higher price does not imply a good performance 12 / 18
  13. 13. at all, as the worst software in our study was also the most expensive one. We concluded ourpaper with a visual approach that can be implemented in software packages to enable interactivedecision making with respect to batching. We believe that our study will offer the decisionmakers with general guidance on software selection and the software companies with a featurethat they can use to enhance the functionality of their products.AcknowledgementWe sincerely thank Necati Alp at Mimag for motivating our study, providing the real world data,and continuously supporting us. We also thank Sevket Atilla at Kaplan Machinery for allowingus to use their equipment’s photograph in our paper.ReferencesGilmore, P. C. and Gomory, R. E. (1961). A linear programming approach to the cutting stock problem. Operations Research, vol: 9, 848--859Kalvelagen, E. (2003). Column Generation with Gams. Gams Development Corp. Washinton DC. 13 / 18
  14. 14. APPENDIX A.Setsi widths /w1*w8/ *different lengths of the steel bars.j pattern/p1*p38/; *patterns in which combination they are going to be used.scalar r raw width /1200/; *bar length is set to 1200.table demanddata(i,*) *demand table for the lengths and their quantities. width demandw1 100 100w2 200 2w3 300 2w4 493 250w5 590 2w6 630 2w7 780 2w8 930 2;table patterndata(j,i) *pattern table with respect to lengths w1 w2 w3 w4 w5 w6 w7 w8p1 2 2 2 0 0 0 0 0p2 4 1 2 0 0 0 0 0p3 5 2 1 0 0 0 0 0 14 / 18
  15. 15. p4 6 0 2 0 0 0 0 0p5 7 1 1 0 0 0 0 0p6 8 2 0 0 0 0 0 0p7 9 0 1 0 0 0 0 0p8 1 0 1 0 0 0 0 0p9 1 2 0 0 0 0 0 0p10 0 2 1 1 0 0 0 0p11 1 0 2 1 0 0 0 0p12 2 1 1 1 0 0 0 0p13 3 2 0 1 0 0 0 0p14 4 0 1 1 0 0 0 0p15 5 1 0 1 0 0 0 0p16 7 0 0 1 0 0 0 0p17 0 0 2 0 1 0 0 0p18 1 1 1 0 1 0 0 0p19 2 2 0 0 1 0 0 0p20 3 0 1 0 1 0 0 0p21 4 1 0 0 1 0 0 0p22 6 0 0 0 1 0 0 0p23 0 1 0 2 0 0 0 0p24 2 0 0 2 0 0 0 0p25 1 0 0 1 1 0 0 0p26 0 0 0 0 2 0 0 0p27 0 2 0 0 0 0 1 0p28 1 0 1 0 0 0 1 0 15 / 18
  16. 16. p29 2 1 0 0 0 0 1 0p30 4 0 0 0 0 0 1 0p31 0 1 0 0 0 0 0 1p32 0 1 1 0 0 1 0 0p33 1 2 0 0 0 1 0 0p34 2 0 0 0 0 0 0 1p35 2 0 1 0 0 1 0 0p36 3 1 0 0 0 1 0 0p37 5 0 0 0 0 1 0 0p38 0 0 0 1 0 1 0 0;parameter d(i); *demand in taken from the table as input.d(i) = demanddata(i, demand);parameter a(i,j); *the amount of the patterns that are going to be used.a(i,j) = patterndata(j,i);integer variable x(j) pattern used;variable z objective;x.up(j) = sum(i, d(i));equationsnumpat number of patterns used(objective)demand(i) meet demand; 16 / 18
  17. 17. numpat.. z=e=sum(j, x(j)); *minimization of the patterns that are going to be useddemand(i).. sum(j, a(i,j)*x(j))=g=d(i); *demand should be satisfied.model cutting /all/;solve cutting using mip minimizing z;display x.l, x.m;Appendix BThe commercial software packages benchmarked in this paper are listed in random order below.The numbering of the software packages within the paper is not with respect to this order. (1D Stock Cutter) (Optimumcut) (The Itemizer 7.0) 17 / 18
  18. 18. (Real Cut 1d) (Pipe Cutting Suite 3.11) (CutLogic) (Plus1D) 18 / 18